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The 1 g voyage: the maths

Special relativity for a ship with constant proper acceleration (what the crew feels), accelerating to halfway and braking the rest. Computed live.

Equations

τ = (2c/g) · arcosh(1 + gD / 2c²)

Time on the ship's clock for a trip of distance D.

T = (2c/g) · sinh(gτ / 2c)

Time on Earth's clock for the same trip.

v_max = c · tanh(gτ / 2c)

Top speed, at the halfway point.

Constants

g	9.80665 m/s² ≈ 1.032 light-years per year²	standard gravity
light-year	9.4607304725808 × 10¹⁵ m	exact

Worked examples ✓ checked on every change

Proxima Centauri (4.24 ly) at 1 g: the crew's time	3.54 years	the formula above
…and Earth's time	5.90 years	the commonly quoted 5.9 years
Centre of the galaxy (26,000 ly): the crew's time	19.8 years	the formula above

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

What's simplified

• No fuel limit: the ship is a perfect photon rocket (the fuel ratio shows what that would still cost).
• No dust, radiation or time to turn round.

Where it breaks

Never, for the kinematics — this is exact special relativity. Only the engineering is imaginary.

Sources: C. Misner, K. Thorne & J. Wheeler, Gravitation (1973), §6; J. Baez, 'The Relativistic Rocket' (Usenet Physics FAQ).